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Universal Chord Theorem

Let $f:[0,1] \to R$ be a real valued continuous function satisfying $f(0)=f(1)$. Then using intermediate value theorem we know for every $n \in N$ there exist two point $a,b \in [0,1]$ at a distance $1/n$ satisfying $f(a)=f(b)$.

Now my question is, for every $r\in [0,1]$ is it possible to find two points $a,b\in [0,1]$ at a distance $r$, satisfying $f(a)=f(b)$ provided $f:[0,1] \to R$ be a real valued continuous function satisfying $f(0)=f(1)$.

as there is a counterexample for $r>1/2$, please consider the case when $r<1/2$.

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marked as duplicate by Martin Sleziak, William, Noah Snyder, Hagen von Eitzen, Henry T. Horton Oct 9 '12 at 21:50

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@martin sorry martin but i was not aware of the universal chord theorem. And thanks for your references. –  timon Sep 20 '12 at 17:34
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1 Answer

up vote 1 down vote accepted

Hint: Consider $r=\frac23$ and $$f(x)=\begin{cases}x&\mathrm{if\ }x\le \frac13\\ 1-2x &\mathrm{if\ } \frac13<x<\frac23\\ x-1 &\mathrm{if\ }x\ge\frac23 \end{cases}$$

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thank you for your answer.I have one doubt. For $r< 1/2$ will we be able to produce counterexample. –  timon Sep 20 '12 at 16:54
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